2020/10/16 by Jason P. Bell, Dragos Ghioca, Bell, Jason +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2010.08579
openalex publication_date 2020/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set X∩Γ when X is a subvariety of a semiabelian variety G over a finite field \mathbbFq and Γ is a finitely generated subgroup of G that is invariant under the q-power Frobenius endomorphism F. That description is here made effective, and extended to arbitrary commutative algebraic groups G and arbitrary finitely generated ℤ[F]-submodules Γ. The approach is to use finite automata to give a concrete description of X∩ Γ. These methods and results have new applications even when specialised to the case when G is an abelian variety over a finite field, X⊆ G a subvariety defined over a function field K, and Γ=G(K). As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in X(K) of bounded height. As an application of the effective description of X∩Γ, decision procedures are given for the following three diophantine problems: Is X(K) nonempty? Is it infinite? Does it contain an infinite coset?